Throughput and Diversity Gain of Buffer-Aided Relaying - IEEE Xplore

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2011 proceedings.

Throughput and Diversity Gain of Buffer-Aided Relaying Nikola Zlatanov† , Robert Schober† , and Petar Popovski†† †

The University of British Columbia,

Abstract— In this paper, we consider a simple network consisting of a source, a half-duplex decode-and-forward relay, and a destination. In contrast to most of the existing literature, we assume that the relay is equipped with a buffer and show that this can lead to substantial performance gains. We propose a simple protocol which chooses either the source-relay or the relay-destination link for transmission depending on the instantaneous channel state information. For this simple protocol, we derive the throughput for adaptive rate transmission and the outage probability for fixed rate transmission. Our results show that, unlike conventional relaying, buffer-aided relaying yields a diversity gain of two. In addition, throughput gains of up to 100 % compared to conventional relaying are possible.

I. I NTRODUCTION The capacity of a three-node transmission systems consisting of a source, a destination, and a single half-duplex decodeand-forward (DF) relay was investigated in [1]. Later, [2] and [3] have sparked renewed interest in relay-assisted communication. Since then the simple three-node system has become a building block for more sophisticated relay networks and has led to the creation of a host of cooperative communication techniques [4]-[7]. It was shown in [1] that the capacity of a three-node DF network without a direct source-destination link is given by the minimum of the source-relay link capacity and the relay-destination link capacity. Furthermore, in [4] the outage probability of this simple network was obtained. In [4]-[7] and most of the existing literature on relaying, it is assumed that relays receive a packet from the source in one time slot and forward it to the destination in the next time slot, which will be referred to as “conventional relaying” in the following. In this paper, we will abandon this paradigm and give relays the freedom to decide in which time slot to receive and in which time slot to transmit. This new approach requires the relays to have buffers. Relays with buffers were also considered in [8] and [9]. In [8], the buffer at the relay was used to enable the relay to receive for a fixed number of time slots before retransmitting the received information in a fixed number of time slots. In [9], relay selection was considered and buffers enable the selection of the relay with the best source-relay channel for reception and the best relay-destination channel for transmission. However, like in conventional relay selection without buffers [10], the source transmits in every other time slot. Thus, the relay selection scheme in [9] achieves a signalto-noise ratio (SNR) gain compared to conventional relay selection but no diversity gain. Both [8] and [9] do not fully exploit the flexibility offered by relays with buffers since the schedule of when the source transmits and when a relay transmits is a priori fixed. In this paper, we assume that one of the three nodes of the network acquires the channel state information (CSI) of the source-relay and relay-destination links and subsequently

††

Aalborg University

exploits this CSI to decide which link is used for transmission in the considered time slot. We propose a link selection criterion and analyze the throughput and the outage probability of the proposed buffer-aided relaying scheme assuming an infinite buffer size. The impact of finite buffer sizes is studied via simulations. We show that buffer-aided relaying achieves significant throughput and diversity gains compared to conventional relaying without buffers since a link is only utilized for transmission when its CSI is good. These gains come at the expense of an additional delay that is introduced by the buffer. Thus, the proposed scheme is most suitable for applications which require high power and bandwidth efficiency but can tolerate delays. The remainder of the paper is organized as follows. In Section II, the considered system model is presented. The optimal link selection threshold is derived in Sections III, and the outage probability is analyzed in Section IV. In Section V, numerical results are presented, and some conclusions are drawn in Section VI. II. S YSTEM M ODEL We consider a three-node communication system comprising a source S, a relay R, and a destination D. The source sends packets to the half-duplex relay, which decodes these packets, possibly stores them in its buffer, and eventually sends them to the destination. We assume that time is divided into slots of equal lengths. The instantaneous link SNRs of the S-R and R-D channels in the ith time slot are denoted by s(i) and r(i), respectively. We assume that the link SNRs are constant during one time slot but change from one time slot to the next due to e.g. the mobility of the involved nodes and/or frequency hopping. s(i) and r(i) are modeled as mutually independent, non-negative random processes with expected values ΩS = E{s(i)} and ΩR = E{r(i)}, respectively, where E{·} denotes expectation. The average link SNRs ΩS and ¯ S and ΩR = γ Ω ¯ R, ΩR can be decomposed as ΩS = γ Ω where γ = PT /σn2 denotes the transmit SNR with PT and σn2 representing the transmit power and the variance of the additive white Gaussian noise at the relay and the destination, ¯ R are the variances of the source¯ S and Ω respectively, and Ω relay and relay-destination channel gains, respectively. Here, we assume that the source and the relay transmit with identical power PT , i.e., power allocation is not considered. We note that in general we do not require s(i) and r(i) to be fully temporally uncorrelated. However, this assumption will be made in some parts of the paper to facilitate analysis. In a given time slot i, one of the nodes acquires knowledge of s(i) and r(i) and makes, based on this knowledge, a decision whether the source or the relay should transmit. Subsequently, this node broadcasts its decision to the other nodes and transmission in time slot i begins. Throughout this

978-1-4244-9268-8/11/$26.00 ©2011 IEEE


paper, we assume that the source node has always data to transmit and the buffer at the relay is not limited in size. The effect of limited buffer size will be investigated in Section V via simulations. In the following, we discuss two different modes of transmission. A. Adaptive Rate Transmission We first consider adaptive transmission, where the source and the relay know s(i) and r(i), respectively, and adjust their transmission rates such that they transmit with the maximum possible rate without causing outages. Source transmits: If the source transmits in a given time slot i, it transmits with rate SSR (i) = log2 (1 + s(i)).

(1)

In the following, for convenience we normalize the number of bits transmitted in one time slot to the number of symbols per time slot. Thus, the data rate is identical to the normalized number of bits per time slot. Hence, the relay receives SSR (i) data bits from the source and appends them to the queue in its buffer. The number of bits in the buffer of the relay at the end of the i-th time slot is denoted by Q(i) and given by Q(i) = Q(i − 1) + SSR (i).

(3)

where we take into account that the maximal number of bits that can be send by the relay is limited by the number of bits in the buffer. The number of data bits remaining in the buffer at the end of time slot i is given by Q(i) = Q(i − 1) − RRD (i),

(4)

which is always non-negative because of (3). Throughput: Because of the half-duplex constraint, we have RRD (i) = 0 and SSR (i) = 0 if the source transmits (and the relay listens) and the relay transmits, respectively. Thus, assuming the source has always data to transmit, the average (normalized) number of bits that arrive at the destination per time slot is given by N 1 RRD (i), N →∞ N i=1

τ = lim

III. A DAPTIVE R ATE T RANSMISSION In this section, we derive a link selection criterion and the corresponding throughput for adaptive rate transmission. A. Problem Formulation Let di ∈ {0, 1} denote a binary decision variable. We set di = 1 if the R-D channel is selected for transmission in time slot i, i.e., the relay transmits and the destination receives. Similarly, we set di = 0 if the S-R channel is selected for transmission in time slot i, i.e., the source transmits and the relay receives. Exploiting di , the number of bits send from the source to the relay and from the relay to the destination in time slot i can be written in compact form as SSR (i) = (1 − di )S(i)

(6)

RRD (i) = di min{R(i), Q(i − 1)},

(7)

(2)

Relay transmits: If the relay transmits in a given time slot, the number of bits transmitted by the relay is given by RRD (i) = min{log2 (1 + r(i)), Q(i − 1)},

contrast, if the relay is selected for transmission in time slot i, we have RRD (i) = min{RR , Q(i − 1)} and SSR (i) = 0. The queue state of the buffer still changes according to (2) and (4). For fixed rate transmission, in time slot i an outage occurs if the source is selected for transmission and log2 (1 + s(i)) < RS or if the relay is selected for transmission and log2 (1 + r(i)) < min{RR , Q(i − 1)}. The outage probability of fixed rate transmission will be derived in Section IV.

(5)

i.e., τ is the throughput of the considered communication system. In Section III, we will introduce a link selection criterion and optimize for maximization of τ . B. Fixed Rate Transmission If the destination selects the link used for transmission, the destination can estimate the CSI of the R-D link and acquire the CSI of the S-R link from the relay through a feedback channel. In this case, the source and the relay are not aware of s(i) and r(i), respectively, and as a consequence, are not able to adapt their transmission rates to the channel state. Thus, the source and the relay transmit with fixed rates RS and RR , respectively. Therefore, if the source is selected for transmission in time slot i, we have SSR (i) = RS and RRD (i) = 0. In

and respectively, where S(i) = log2 (1+s(i)) and R(i) = log2 (1+ r(i)). Consequently, the throughput in (5) can be rewritten as N 1 di min{R(i), Q(i − 1)}. N →∞ N i=1

τ = lim

(8)

In order to limit complexity, we consider link selection criteria which only exploit the CSI in the considered time slot, i.e., 1 if F(r(i)) ≥ ρF(s(i)) di = (9) 0 otherwise where ρ and F(·) are referred to as decision threshold and decision function, respectively. The decision function is a nonnegative, smooth, and increasing function, i.e., F(x + ) > F(x) for > 0. We assume that the inverse of F(·) exists and denote it by F −1 (·). In order to maximize the throughput in (8) for a given decision function F(·), ρ has to be optimized, cf. Section III-B. We note that as will be explained later, if ρ is optimized, the performance of the link selection criterion in (9) cannot be improved by also taking into account the queue state for link selection, cf. Remark 1. On the other hand, the choice of the decision function influences the achievable throughput and the computational complexity required to find the optimal ρ, cf. Sections III-C, and V. B. Optimal ρ for General F(·) Let us first define the mean arrival rate, A, and the mean departure rate, D, of the queue in the buffer of the relay as [11] N 1 (1 − di )S(i) N →∞ N i=1

A = lim

(10)


and D = lim

N →∞

1 N

N

di min{R(i), Q(i − 1)},

(11)

i=1

respectively. We note that the departure rate of the queue is equal to the throughput. The queue is said to be unstable or absorbing if A > D = τ . The following theorem characterizes the optimal ρ in terms of the state of the queue in the buffer of the relay. Theorem 1: For a given decision function F(·), the optimal ρ, ρopt , that maximizes the throughput is found at the point where the queue in the buffer of the relay switches from an absorbing to a non-absorbing queue, i.e., the queue is nonabsorbing for ρ = ρopt − and absorbing for ρ = ρopt + , where → 0+ denotes an arbitrarily small positive number. Proof: From (9) and (10), we observe that A is an increasing function of ρ. As ρ increases, the number of time slots with di = 0 will increase and consequently A will increase. Furthermore, because of the law of the conservation of flow, A ≥ τ is valid and equality holds if and only if the queue is non-absorbing. If we set ρ = ∞, we have di = 0 for all time slots. Thus, A assumes its maximum value and τ assumes its minimum value τ = 0. The queue at the relay absorbs all bits transmitted by the source. By gradually decreasing ρ, we decrease A, increase τ , and decrease the amount of data that gets absorbed by the queue. When we reach ρ = ρopt for which A = τ |ρ=ρopt but A > τρ=ρ|opt + , → 0+ , the throughput τ is maximized. This is exactly the point when the queue at the relay switches from an absorbing queue to a non-absorbing queue. If we decrease ρ further, A = τ is still valid and the queue is non-absorbing but A has a smaller value. This completes the proof. Theorem 2: Assuming non-negative, ergodic, and stationary random processes s(i) and r(i), for a given decision function F(·), the optimal ρ, ρopt , that maximizes the throughput is the solution to E{(1 − di )S(i)} = E{di R(i)},

(12)

where di is defined in (9). Proof: We know from Theorem 1 that A = τ holds for ρ = ρopt . Let us denote the set of indices i with R(i) ≤ Q(i− ¯ 1) by C and the set of indices i with R(i) > Q(i − 1) by C. ¯ ¯ Let cardinalities of C and C be M = |C| and N − M = |C|, respectively. If the queue is absorbing, on average the number of bits that are arriving at the relay exceeds the number of bits leaving the relay. Thus, R(i) ≤ Q(i − 1) holds almost always and M/N → 0. Consequently, the throughput in (8) simplifies to 1 1 di R(i) + lim di Q(i − 1) τ = lim N →∞ N N →∞ N ¯ i∈C

i∈C

N −M = lim E{di R(i)} = E{di R(i)}, (13) N →∞ N where we have exploited the ergodicity of r(i). Since according to Theorem 1, the queue is absorbing for ρ = ρopt + , → 0+ , and F(·) is a smooth function, (13) is also valid for ρ = ρopt . Hence, (12) follows directly from Theorem 1, (13), and the assumption that s(i) is ergodic. This completes the proof.

Remark 1: According to Theorem 2, for the optimal value of ρ, the queue is non-absorbing but is almost always filled to such a level that the number of bits in the queue exceed the number of bit that can be transmitted over the relay-destination channel. Thus, for the optimal value of ρ, the performance of the link selection criterion in (9) cannot be improved by also taking into account the state of the queue at the relay. We are now ready to provide a formula that can be used for computation of the optimal ρ. Lemma 1: Denote the probability density functions (pdfs) of s(i) and r(i) by fs (s(i)) and fr (r(i)), respectively. Then, the optimal ρ is the solution of

∞

0

G(r)

−

∞

∞

0

log2 (1 + s)fs (s)ds fr (r)dr

∞

H(s)

log2 (1 + r)fr (r)dr fs (s)ds = 0

(14)

where the integral limits are given by G(r) = F −1 (F(r)/ρ) and H(s) = F −1 (ρF(s)). Proof: The proof follows directly from Theorem 2 and basic probability theory. C. Throughput in Rayleigh Fading In this subsection, we assume Rayleigh fading, i.e., the link SNRs s(i) and r(i) are exponentially distributed with means ΩS and ΩR , respectively. Their pdfs are given by fs (s) = e−s/ΩS /ΩS and fr (r) = e−r/ΩR /ΩR . In the following, we will determine ρopt and the corresponding maximum throughput τmax for two different decision functions F(·). 1) F(x) = x: In this case, the limits G(r) and H(s) in (14) are given by G(r) = r/ρ, and H(s) = ρs. Thus, after some elementary manipulations, (14) simplifies to 1 1 1 1 Ei − exp Ei (15) ΩS ΩS Ω ΩR R ΩR ΩR + ρΩS ΩR + ρΩS − Ei exp ΩR + ρΩS ΩS ΩR ΩS ΩR ρΩS ΩR + ρΩS ΩR + ρΩS + Ei =0 exp ΩR + ρΩS ρΩS ΩR ρΩS ΩR

exp

∞ where Ei (x) = x e−t /t dt, x > 0, denotes the exponential integral function. The optimal value of ρ for F(x) = x, ρopt,1 , can be obtained via a simple one dimensional search from (15). The corresponding maximal throughput can be obtained from τmax,1 = E{d log2 (1 + r)} and can be computed as 1 1 1 exp Ei (16) ln(2) ΩR ΩR 1 ΩR + ρΩS ΩR + ρΩS ρΩS − Ei , exp ln(2) ΩR + ρΩS ρΩS ΩR ρΩS ΩR

τmax,1

=

where ρ = ρopt,1 . Remark 2: If ΩS = ΩR , the solution to (15) is ρopt,1 = 1.


2) F(x) = log2 (1 + x): In this case, we have G(r) = 1 (1 + r) ρ − 1 and H(s) = (1 + s)ρ − 1 in (14). Thus, after some manipulations, we obtain from (14) 1 ∞

1 (r + 1) ρ − 1 (17) exp − ln (r + 1) ρ ΩS 0 1 ρ 1 1 r (r + 1) +e ΩS Ei dr × exp − ΩS ΩR ΩR ∞

(s + 1)ρ − 1 − exp − ln ((s + 1)ρ ) ΩR 0 1 (s + 1)ρ 1 s +e ΩR Ei × ds = 0. exp − ΩR ΩS ΩS The optimal ρ, ρopt,2 , can be found numerically from (17). The corresponding maximum throughput is obtained as ∞

(s + 1)ρ − 1 1 exp − τmax,2 = ln(2) 0 Ω R 1 (s + 1)ρ ρ ΩR × ln ((s + 1) ) + e Ei ΩR 1 s × ds, (18) exp − ΩS ΩS where ρ = ρopt,2 . Remark 3: If ΩS = ΩR then the solution to (17) is ρopt,2 = 1. Remark 4: Since for the second decision function, F(x) = log2 (1+x), the channel capacity is used as selection criterion, τmax,2 ≥ τmax,1 holds. In fact, τmax,2 is the maximum achievable throughput for any decision function. However, the achievable gain compared to the first decision function, F(x) = x, is small, cf. Fig. 1, and the search complexity for the optimal ρ is higher since (17) requires numerical integration. IV. F IXED R ATE T RANSMISSION For fixed rate transmission, for simplicity, we only consider the decision function F(x) = x. The link is still selected according to (9). We assume that the relay adds the decoded bits to its queue regardless of whether or not they have been decoded correctly. A. Optimal Decision Threshold The following theorem provides the optimal decision threshold, ρopt,3 . Theorem 3: Consider Rayleigh fading links and assume the source and the relay transmit with rates RS and RR , respectively. In this case, for decision function F(x) = x, the optimal decision threshold maximizing the throughput is given by ρopt,3 = ΩR RR /(ΩS RS ).

(19)

Proof: Using similar arguments as in the proof of Theorem 1 for the adaptive transmission case, it can be shown that the throughput is maximized when the queue switches from an absorbing queue to a non-absorbing queue. Using

then similar arguments as in the proof of Theorem 2, we find that the optimal ρ is the solution to RS (1 − E{di }) − E{di }RR = 0.

(20)

On the other hand, we obtain from (9) E{di } = Pr{r(i) > ρs(i)} = ΩR /(ΩR + ρΩS ). Combining this with (20) leads directly to (12), which completes the proof. Lemma 2: The maximum throughput in the fixed rate case is given by τmax,3 = RS RR /(RS + RR )

(21)

independent of ΩS and ΩR . Proof: The maximum throughput is given by τmax,3 = E{di }RR |ρ=ρopt,3 . Using now the expression for E{di } given in the proof of Theorem 3 and Theorem 3 itself, we obtain (21). Remark 5: Although the maximum throughput for fixed rate transmission does not depend on ΩS and ΩR , the corresponding outage probability does. B. Outage Probability For fixed rate transmission, outages are unavoidable. Thus, the outage probability is a relevant performance metric. To facilitate our analysis, we assume in this subsection that s(i) and r(i) are temporally uncorrelated. Theorem 4: Consider a buffer-aided relaying system where the source and the relay transmit with fixed rate, the decision function is F(x) = x is used to select one of the links for transmission, and the source–relay and relay-destination channels are affected by temporally uncorrelated Rayleigh fading. In this case, the outage probability decays on a log-log scale with a slope of two with increasing transmit SNR γ, i.e., the diversity order is two. Proof: An outage event occurs if the source transmits in time slot i but the source-relay link is in outage or if the source-relay link was not in outage in time slot i but the relaydestination link is in outage when the packet is retransmitted by the relay in time slot j > i. For simplicity, we use the notation s = s(i), r = r(i), r¯ = r(j), and s¯ = s(j). Note that r, s, r¯, and s¯ are mutually independent. Thus, the outage probability can be expressed as Pout = Pr {[log2 (1 + s) < RS |ρs > r] OR ([log2 (1 + s) > RS |ρs > r] AND r > ρ¯ s])} [log2 (1 + r¯) < min{RR , Q(j − 1)}|¯ Pr {log2 (1 + s) < RS AND ρs > r} = Pr{ρs > r} s} Pr {log2 (1 + r¯) < min{RR , Q(j − 1)} AND r¯ > ρ¯ + Pr{¯ r > ρ¯ s} Pr {log2 (1 + s) < RS AND ρs > r} − Pr{ρs > r} s} Pr {log2 (1 + r¯) < min{RR , Q(j − 1)} AND r¯ > ρ¯ × Pr{¯ r > ρ¯ s} = P1 + (1 − P1 ) Pr ρ¯ s < r¯ < 2min{RR ,Q(j−1)} − 1 , (22) × Pr{¯ r > ρ¯ s}


where P1 is given by P1 = Pr r/ρ < s < 2RS − 1 /Pr{s > r/ρ} ΩR ΩR + ρΩS RS = 1+ exp − (2 − 1) ρΩS ΩS ΩR RS ΩR + ρΩS 2 −1 . (23) − exp − ρΩS ΩS Since Q(j − 1) is unknown, we cannot obtain the exact outage probability. However, exploiting the fact that P 1 < min{RR ,Q(j−1)} − 1 ≤ 1 and the inequality Pr ρ¯ s < r ¯ < 2 Pr ρ¯ s < r¯ < 2RR − 1 , we obtain from (22) the upper bound Pout ≤ P1 + P2 − P1 P2 , where

(24)

Pr ρ¯ s < r¯ < 2RR − 1 /Pr {¯ r > ρ¯ s} ρΩS ΩR + ρΩS RR = 1+ exp − (2 − 1) ΩR ρΩS ΩR RR 2 −1 ΩR + ρΩS (25) exp − − ΩR ΩR

Clearly, buffer-aided relaying leads to substantial throughput gains compared to conventional relaying. Both considered decision functions lead to a very similar performance, although at very high ratios ΩR /ΩS the optimal decision function F(x) = log2 (1 + x) yields a small throughput gain. The ratio τmax /τconv approaches two as ΩR /ΩS → 0 and ΩR /ΩS → ∞. For ΩR /ΩS → 0, buffer-aided relaying uses the sourcerelay link very rarely since comparatively large amounts of data can be transferred to the relay in a single time slot. Thus, the relay can almost always transmit as compared to half of the time in conventional relaying. On the other hand, for ΩR /ΩS → ∞, it is the relay-destination channel that is used very rarely and the source can transmit almost all the time, which results in twice the throughput as for conventional relaying.

B. Outage Probability For a fair comparison of the outage probabilities of bufferaided relaying with decision function F(x) = x and conventional relaying we require both schemes to have the same throughput. This is achieved by choosing RS = RR = R0 for both relaying schemes. For buffer-aided relaying the decision threshold is chosen as specified in Theorem 3. The ¯ ¯ Exploiting now ΩS = γ ΩS and ΩR = γ ΩR and the Taylor outage probabilities of buffer-aided and conventional relaying series expansion e−t = 1−t+t2 /2, we obtain for high transmit as functions of the transmit SNR γ are shown in Fig. 2 for SNRs (i.e., γ → ∞) from (24) the asymptotic bound Pout ≤ ¯ ¯ S . The ΩR = 1, R0 = 2 b/s/Hz, and different values of Ω RS 2 RR 2 ¯ ¯ ¯ ¯ outage probability of conventional relaying was calculated (ΩR /ΩS + ρ)(2 − 1) + (1/ρ + ΩS /ΩR )(2 − 1) . (26) based on the results in [4]. For the outage probability of ¯SΩ ¯ Rγ2 2Ω buffer-aided relaying, we show the upper bound in (24) and From (26), we directly observe that Pout decays with slope computer simulations. As predicted in Remark 6, the upper two on a log-log scale for high SNR γ, which concludes the bound is tight since the optimal ρ was used and the event proof. Q(j − 1) < RR has a negligible effect. Furthermore, as Remark 6: The tightness of the upper bound in (24) de- expected from Theorem 4, buffer-aided relaying achieves a pends on the choice of ρ. For ρ ≥ ρopt,3 , the queue in the diversity order of two, whereas conventional relaying achieves buffer is absorbing or at the boundary between the absorbing only a diversity order of one, which underlines the superiority and non-absorbing state and Q(j − 1) < RR will occur with of buffer-aided relaying. negligible probability. Thus, in this case, the upper bound in (24) is tight. C. Limited Buffer Size Remark 7: The diversity order of conventional, non-bufferSo far, we have assumed that the size of the buffer at aided DF relaying is equal to one [4]. This means buffer-aided the relay is infinite. Clearly, in practice, this will not be the relaying achieves a diversity gain compared to conventional case. Thus, we investigate in Fig. 3 the effect of a limited relaying. buffer size L on the outage probability. Here, we assume that one packet is transmitted in each time slot and L denotes V. N UMERICAL AND S IMULATION R ESULTS the number of packets that can be stored at the relay. We In this section, we evaluate the performance of buffer-aided ¯S = Ω ¯ R = 1. assume RS = RR = R0 = 2 b/s/Hz and Ω DF relaying and compare it with that of conventional relaying. The scheme with finite buffer size operates exactly the same Throughout this section, we assume temporally independent way as the scheme with infinite buffer size except when the Rayleigh fading. buffer is empty (full) the source (relay) is forced to transmit in the next time slot. In addition to the simulations for finite A. Throughput for Adaptive Rate Transmission In Fig. 1, we show the ratio of the optimal through- buffer size, we also show analytical results for buffer-aided put of buffer-aided relaying, τmax , and the throughput of relaying with infinite buffer size and conventional relaying. conventional relaying, τconv , which is given by τconv = For all buffer-aided schemes the decision function F(x) = x is adopted. As expected, as L increases, buffer-aided relaying −1 −1 −1 min{exp(Ω−1 S )Ei (ΩS ), exp(ΩR )Ei (ΩR )}/(2 loge (2)), as a function of ΩR /ΩS for several different values of ΩS . with finite buffer size approaches the performance of bufferFor buffer-aided relaying, we considered decision functions aided relaying with infinite buffer size. Note that the maximum F(x) = log2 (1 + x) and F(x) = x and calculated the cor- delay of the considered system is equal to the buffer size. responding throughputs based on (18) and (16), respectively.1 Therefore, even for moderate delays of 30 packets, significant performance gains compared to conventional relaying are 1 The results in Fig. 1 have been confirmed by computer simulations. These possible. However, for any finite buffer size, the slope of the simulations are not shown here for clarity of presentation. outage probability curves will approach unity for sufficiently P2

=


0

10

Ω =0.1

ΩS=1

τ max/τ conv

S

1.8

Policy: F(x) = x Policy: F(x) = log2 (1 + x)

1.6 1.4

Ω =20 S

0

1

2

3

4 5 Ω /Ω R

6

L=5; 10; 30; 100; 500 −2

10

−4

7

8

9

S

Ratio τmax /τconv vs. ΩR /ΩS for several different values of ΩS .

Fig. 1.

Outage Probability

2

10

0

Conventional Relaying BA Relaying (Infinite Buffer) BA Relaying (Finite Buffer)

5

10

15 20 γ (in dB)

25

30

35

Fig. 3. Outage probability of buffer-aided (BA) relaying with limited buffer ¯S = Ω ¯ R = 1. size L. RS = RR = R0 = 2 b/s/Hz and Ω

0

Outage Probability

10

Ω¯ S = 0.1; 1; 10

−5

10

Conventional Relaying BA Relaying (Upper Bound) BA Relaying (Simulation)

0

10

20 30 γ (in dB)

R EFERENCES

Ω¯ S = 0.1; 1; 10

40

50

Fig. 2. Outage probability of buffer-aided (BA) and conventional relaying ¯ R = 1. vs. γ. RS = RR = R0 = 2 b/s/Hz and Ω

high SNR. This behavior is caused by empty and full buffers, where the source or the relay have to transmit regardless of the quality of their link. VI. C ONCLUSIONS In this paper, we proposed a novel relaying protocol for relays with buffers. In contrast to conventional relaying, where the source and the relay transmit in successive time slots regardless of the channel state, in the proposed scheme, always the node with the stronger link is selected for transmission. We derived the throughput for adaptive rate transmission and the outage probability for fixed rate transmission. Furthermore, we could show that the diversity order of the proposed bufferaided relaying scheme is two, which results in significant performance gains compared to conventional relaying. Thus, buffers seem an efficient means to improve the performance of relay networks for non-delay limited applications. Interesting topics for future work include using the considered simple three-node network as a building block for larger networks, as has been done for relays without buffers before, and the analysis of buffer-aided relaying with buffers of finite size.

[1] T. Cover, A.E. Gamal, ”Capacity Theorems for the Relay Channel” IEEE Trans. Inf. Theory, vol. 25, pp. 572 - 584, Sep. 1979. [2] A. Sendonaris, E. Erkip, and B. Aazhang, ”User Cooperation Diversity. Part I. System Description”, IEEE Trans. Commun., vol. 51, pp. 19271938, Nov. 2003. [3] A. Sendonaris, E. Erkip, and B. Aazhang, ”User Cooperation Diversity. Part II. Implementation Aspects and Performance Analysis”, IEEE Trans. Commun., vol. 51, pp. 1939-1948, Nov. 2003. [4] J. N. Laneman, D.N.C. Tse, and G. W. Wornell, ”Cooperative Diversity in Wireless Networks: Efficient Protocols and Outage Behavior”, IEEE Trans. Inf. Theory, vol. 50, pp. 3062-3080, Dec. 2004. [5] J. N. Laneman and G. W. Wornell, ”Distributed Space Time-Coded Protocols for Exploiting Cooperative Diversity in Wireless Networks”, IEEE Trans. Inf. Theory, vol. 49, pp. 2415-2425, Oct. 2003. [6] A. Nosratinia, T.E. Hunter, and A. Hedayat, ”Cooperative Communication in Wireless Networks”, IEEE Commun. Magaz., vol. 42, pp. 74-80, Oct. 2004. [7] R.U. Nabar, H. B¨olcskei, F.W. Kneub¨uhler, ”Fading Relay Channels: Performance Limits and Space-Time Signal Design”, IEEE J. Selec. Areas Comm., vol. 22, pp. 1099-1109 , Aug. 2004. [8] B. Xia, Y. Fan, J. Thompson, and H. V. Poor, ”Buffering in a Three-Node Relay Network”, IEEE Trans. Wireless Commun., vol. 7, pp. 4492-4496, Nov. 2008. [9] A. Ikhlef, D. Michalopoulos, and R. Schober ”Buffers Improve the Performance of Relay Selection”, Submitted to IEEE Global Telecommun. Conf. (Globecom), Mar. 2011. [Online] http://ece.ubc.ca/∼dio/Globecom11.pdf [10] A. Bletsas, A. Khisti, D. Reed, and A. Lippman, ”A Simple Cooperative Diversity Method Based on Network Path selection”, IEEE J. Select. Areas Commun., vol. 24, pp. 659-672, Mar. 2006. [11] Leonidas Georgiadis, Michael J. Neely and Leandros Tassiulas, ”Resource Allocation and Cross Layer Control in Wireless Networks”, Now Publishers Inc, 2006.